Time-scale calculus

Abstract: An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.

Abstract: Preface. Acknowledgments. Special Functions Of Preface. Acknowledgements. Special Functions of the Fractional Calculus. Gamma Function. Mittag-Leffler Function. Wright Function. Fractional Derivatives and Integrals. The Name of the Game. Grunwald-Letnikov Fractional Derivatives. Riemann-Liouville Fractional Derivatives. Some Other Approaches. Sequential Fractional Derivatives. Left and Right Fractional Derivatives. Properties of Fractional Derivatives. Laplace Transforms of Fractional Derivatives. Fourier Transforms of Fractional Derivatives. Mellin Transforms of Fractional Derivatives. Existence and Uniqueness Theorems. Linear Fractional Differential Equations. Fractional Differential Equation of a General Form. Existence and Uniqueness Theorem as a Method of Solution. Dependence of a Solution on Initial Conditions. The Laplace Transform Method. Standard Fractional Differential Equations. Sequential Fractional Differential Equations. Fractional Green's Function. Definition and Some Properties. One-Term Equation. Two-Term Equation. Three-Term Equation. Four-Term Equation. Calculation of Heat Load Intensity Change in Blast Furnace Walls. Finite-Part Integrals and Fractional Derivatives. General Case: n-term Equation. Other Methods for the Solution of Fractional-order Equations. The Mellin Transform Method. Power Series Method. Babenko's Symbolic Calculus Method. Method of Orthogonal Polynomials. Numerical Evaluation of Fractional Derivatives. Approximation of Fractional Derivatives. The "Short-Memory" Principle. Order of Approximation. Computation of Coefficients. Higher-order Approximations. Numerical Solution of Fractional Differential Equations. Initial Conditions: Which Problem to Solve? Numerical Solution. Examples of Numerical Solutions. The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations. Fractional-Order Systems and Controllers. Fractional-Order Systems and Fractional-Order Controllers. Example. On Viscoelasticity. Bode's Analysis of Feedback Amplifiers. Fractional Capacitor Theory. Electrical Circuits. Electroanalytical Chemistry. Electrode-Electrolyte Interface. Fractional Multipoles. Biology. Fractional Diffusion Equations. Control Theory. Fitting of Experimental Data. The "Fractional-Order" Physics? Bibliography. Tables of Fractional Derivatives. Index.

Abstract: Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

Abstract: The paper begins with a review of the algebras related to Kronecker products. These algebras have several applications in system theory including the analysis of stochastic steady state. The calculus of matrix valued functions of matrices is reviewed in the second part of the paper. This calculus is then used to develop an interesting new method for the identifiication of parameters of lnear time-invariant system models.